Extremal rays of non-integral $L$-length

Let $X$ be a smooth complex projective variety and let $L$ be a line bundle on it. We describe the structure of the pre-polarized manifold $(X,L)$ for non integral values of the invariant ${\tau }_L(R):=-K_X\cdot \mathrm{\Gamma }\mathrm{/}(L\cdot \mathrm{\Gamma })$, where $\mathrm{\Gamma }$ is a minimal curve of an extremal ray $R:={\mathbb{R}}_{+}[\mathrm{\Gamma }]$ on $X$ such that $L\cdot R>0$.